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Difference between revisions of "2005 PMWC Problems"

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The following is a list of [[PMWC]] problems from the year 2005
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== Problem I1 ==
 
== Problem I1 ==
 
What is the greatest possible number one can get by discarding <math>100</math> digits, in any order, from the number <math>123456789101112 \dots 585960</math>?
 
What is the greatest possible number one can get by discarding <math>100</math> digits, in any order, from the number <math>123456789101112 \dots 585960</math>?
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== Problem I2 ==
 
== Problem I2 ==
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Let <math>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}</math>, where <math>a</math> and <math>b</math> are different four-digit positive integers (natural numbers) and <math>c</math> is a five-digit positive integer (natural number).  What is the number <math>c</math>?
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Let <math>\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}</math>, where <math>a</math> and <math>b</math> are different four-digit positive integers (natural numbers) and <math>c</math> is a five-digit positive integer (natural number)ems/Problem I2|Solution]]
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−
[[2005 PMWC Problems/Problem I2|Solution]]
 
  
 
== Problem I3 ==
 
== Problem I3 ==
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== Problem I4 ==
 
== Problem I4 ==
 
The larger circle has radius 12 cm. Each of the six identical smaller circles
 
The larger circle has radius 12 cm. Each of the six identical smaller circles
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touches its two neighbours and the larger circle. What is the radius of the
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touches its two neighbors and the larger circle. What is the radius of the
−
smaller circle?
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smaller circles?
  
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{{image}}
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<asy>
 +
unitsize(0.5cm);
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draw((0,3)..(3,0)..(0,-3)..(-3,0)..cycle);
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for (int i=0;i<6;i=i+1){
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draw(dir(60*i)..3*dir(60*i)..cycle);
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}
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</asy>
  
 
[[2005 PMWC Problems/Problem I4|Solution]]
 
[[2005 PMWC Problems/Problem I4|Solution]]
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segment <math>ED</math>, <math>BCA</math> is a right angled triangle, <math>AC</math> is perpendicular to <math>BC</math>.Suppose that <math>BC=8cm</math>, <math>AC=7cm</math>, and the area of the shaded regions is <math>12cm^2</math> more than that of the triangle <math>AFG</math>. What is the length of <math>CG</math>?
 
segment <math>ED</math>, <math>BCA</math> is a right angled triangle, <math>AC</math> is perpendicular to <math>BC</math>.Suppose that <math>BC=8cm</math>, <math>AC=7cm</math>, and the area of the shaded regions is <math>12cm^2</math> more than that of the triangle <math>AFG</math>. What is the length of <math>CG</math>?
  
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{{image}}
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<asy>
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fill((0,0)--(1,1)--(3,1)--(2,0)--cycle,grey);
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fill((0,0)--(2,1.5)--(2,0)--cycle,white);
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draw((0,0)--(2,1.5)--(2,0)--cycle,dot);
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draw((0,0)--(1,1)--(3,1)--(2,0)--cycle,dot);
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MP("B",(0,0),W);MP("A",(2,1.5),N);MP("C",(2,0),S);
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MP("E",(1,1),NW);MP("D",(3,1),NE);
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MP("F",(1.5,1),NW);MP("G",(2,1),NE);
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draw((1.9,0)--(1.9,.1)--(2,.1),linewidth(.75));
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</asy>
  
 
[[2005 PMWC Problems/Problem I12|Solution]]
 
[[2005 PMWC Problems/Problem I12|Solution]]
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1+2 &=& 3 \\
 
1+2 &=& 3 \\
 
4+5+6 &=& 7+8 \\
 
4+5+6 &=& 7+8 \\
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9+10+11+12 &=& 13+14+15</cmath>
+
9+10+11+12 &=& 13+14+15\end{eqnarray*}</cmath>
 
<cmath>\vdots</cmath>
 
<cmath>\vdots</cmath>
 
If this pattern is continued, find the last number in the <math>80</math>th row
 
If this pattern is continued, find the last number in the <math>80</math>th row
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== Problem T9 ==
 
== Problem T9 ==
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Select 8 of the 9 given numbers: 2, 3, 4, 7, 10, 11, 12, 13, 15 and
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place them in  the vacant squares so  that the average of  the
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numbers in  each  row and  column  is the same. Complete the
 +
following table.
 +
 +
<math>\begin{tabular}{ |c | c | c | c |}
 +
    \hline
 +
    1 &  &  &  \\ \hline
 +
      & 9 & & 5 \\ \hline
 +
      & & 14 &  \\ \hline
 +
  \end{tabular}</math>
  
  

Latest revision as of 19:49, 2 October 2020

The following is a list of PMWC problems from the year 2005

Problem I1

What is the greatest possible number one can get by discarding $100$ digits, in any order, from the number $123456789101112 \dots 585960$?

Solution

Problem I2

Let $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{2005}$, where $a$ and $b$ are different four-digit positive integers (natural numbers) and $c$ is a five-digit positive integer (natural number)ems/Problem I2|Solution]]

Problem I3

Let $x$ be a fraction between $\frac{35}{36}$ and $\frac{91}{183}$. If the denominator of $x$ is $455$ and the numerator and denominator have no common factor except $1$, how many possible values are there for $x$?

Solution

Problem I4

The larger circle has radius 12 cm. Each of the six identical smaller circles touches its two neighbors and the larger circle. What is the radius of the smaller circles?

[asy] unitsize(0.5cm); draw((0,3)..(3,0)..(0,-3)..(-3,0)..cycle); for (int i=0;i<6;i=i+1){ draw(dir(60*i)..3*dir(60*i)..cycle); } [/asy]

Solution

Problem I5

Consider the following conditions on the positive integer (natural number) $a$:

1. $3a + 5 > 40$

2. $49a \ge 301$

3. $20a \le 999$

4. $101a + 53 \ge 2332$

5. $15a-7 \ge 144$

If only three of these conditions are true, what is the value of $a$?

Solution

Problem I6

A group of $100$ people consists of men, women and children (at least one of each). Exactly $200$ apples are distributed in such a way that each man gets $6$ apples, each woman gets $4$ apples and each child gets $1$ apple. In how many possible ways can this be done?

Solution

Problem I7

How many numbers are there in the list $1, 2, 3, 4, 5, \dots, 10000$ which contain exactly two consecutive $9$'s such as $993, 1992$ and $9929$, but not $9295$ or $1999$?

Solution

Problem I8

Some people in Hong Kong express $2/8$ as 8th Feb and others express $2/8$ as 2nd Aug. This can be confusing as when we see $2/8$, we don’t know whether it is 8th Feb or 2nd Aug. However, it is easy to understand $9/22$ or $22/9$ as 22nd Sept, because there are only $12$ months in a year. How many dates in a year can cause this confusion?

Solution

Problem I9

There are four consecutive positive integers (natural numbers) less than $2005$ such that the first (smallest) number is a multiple of $5$, the second number is a multiple of $7$, the third number is a multiple of $9$ and the last number is a multiple of $11$. What is the first of these four numbers?

Solution

Problem I10

A long string is folded in half eight times, then cut in the middle. How many pieces are obtained?

Solution

Problem I11

There are 4 men: A, B, C and D. Each has a son. The four sons are asked to enter a dark room. Then A, B, C and D enter the dark room, and each of them walks out with just one child. If none of them comes out with his own son, in how many ways can this happen?

Solution

Problem I12

In the figure below, $BCDE$ is a parallelogram, points $F$ and $G$ are on the segment $ED$, $BCA$ is a right angled triangle, $AC$ is perpendicular to $BC$.Suppose that $BC=8cm$, $AC=7cm$, and the area of the shaded regions is $12cm^2$ more than that of the triangle $AFG$. What is the length of $CG$?

[asy] fill((0,0)--(1,1)--(3,1)--(2,0)--cycle,grey); fill((0,0)--(2,1.5)--(2,0)--cycle,white); draw((0,0)--(2,1.5)--(2,0)--cycle,dot); draw((0,0)--(1,1)--(3,1)--(2,0)--cycle,dot); MP("B",(0,0),W);MP("A",(2,1.5),N);MP("C",(2,0),S); MP("E",(1,1),NW);MP("D",(3,1),NE); MP("F",(1.5,1),NW);MP("G",(2,1),NE); draw((1.9,0)--(1.9,.1)--(2,.1),linewidth(.75)); [/asy]

Solution

Problem I13

Sixty meters of rope is used to make three sides of a rectangular camping area with a long wall used as the other side. The length of each side of the rectangle is a natural number. What is the largest area that can be enclosed by the rope and the wall?

Solution

Problem I14

On a balance scale, three green balls balance six blue balls, two yellow balls balance five blue balls and six blue balls balance four white balls. How many blue balls are needed to balance four green, two yellow and two white balls?

Solution

Problem I15

The sum of the two three-digit integers, $\text{6A2}$ and $\text{B34}$, is divisible by $18$. What is the largest possible product of $\text{A}$ and $\text{B}$?

Solution

Problem T1

Call an integer "happy", if the sum of its digits is $10$. How many "happy" integers are there between $100$ and $1000$?

Solution

Problem T2

Compute the sum of $a$, $b$ and $c$ given that $\frac{a}{2}=\frac{b}{3}=\frac{c}{5}$ and the product of $a$, $b$ and $c$ is $1920$.

Solution

Problem T3

Replace the letters $a$, $b$, $c$ and $d$ in the following expression with the numbers $1$, $2$, $3$ and $4$, without repetition: \[a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d}}}\] Find the difference between the maximum value and the minimum value of the expression.

Solution

Problem T4

Buses from town A to town B leave every hour on the hour (for example: 6:00, 7:00, …). Buses from town B to town A leave every hour on the half hour (for example: 6:30, 7:30, …). The trip between town A and town B takes 5 hours. Assume the buses travel on the same road. If you get on a bus from town A, how many buses from town B do you pass on the road (not including those at the stations)?

Solution

Problem T5

Mr. Wong has a $7$-digit phone number $\text{ABCDEFG}$. The sum of the number formed by the first $4$ digits $\text{ABCD}$ and the number formed by the last $3$ digits $\text{EFG}$ is $9063$. The sum of the number formed by the first $3$ digits $\text{ABC}$ and the number formed by the last $4$ digits $\text{DEFG}$ is $2529$. What is Mr. Wong’s phone number?

Solution

Problem T6

\begin{eqnarray*} 1+2 &=& 3 \\ 4+5+6 &=& 7+8 \\ 9+10+11+12 &=& 13+14+15\end{eqnarray*} \[\vdots\] If this pattern is continued, find the last number in the $80$th row (e.g. the last number of the third row is $15$).

Solution

Problem T7

Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a 2-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using $\pi=3.14$ or $\pi=22/7$.

Solution

Problem T8

An isosceles right triangle is removed from each corner of a square piece of paper so that a rectangle of unequal sides remains. If the sum of the areas of the cut-off pieces is $200 \text{cm}^2$ and the lengths of the legs of the triangles cut off are integers, find the area of the rectangle.

Solution

Problem T9

Select 8 of the 9 given numbers: 2, 3, 4, 7, 10, 11, 12, 13, 15 and place them in the vacant squares so that the average of the numbers in each row and column is the same. Complete the following table.

$\begin{tabular}{ |c | c | c | c |} \hline 1 & & & \\ \hline & 9 & & 5 \\ \hline & & 14 & \\ \hline \end{tabular}$


Solution

Problem T10

Find the largest 12-digit number for which every two consecutive digits form a distinct 2-digit prime number.

Solution

See Also

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